Piezoelectric resonators are widely used in electric circuits because, not only are they small and light, but also they are more stable in frequency and suffer less change over time than other electric parts. Among piezoelectric resonators, high-frequency crystal resonators, in which a cavity is formed in a part of a crystal substrate, are currently used especially for VHF and UHF bands.
FIG. 3(a) shows a plan drawing of a conventional high-frequency quartz resonator 30 and FIG. 3(b) is a cross section figure of the conventional high-frequency quarts resonator 30 at the 3(b)–3(b) plane. A cavity 32 is formed by photolithography and etching in the center of one of the main facets of an AT-cut crystal substrate 31 wherein the cavity 32 is a resonator. An electrode 33a is formed in the flat side of the crystal substrate 31, and a lead electrode 34a extends from the electrode 33a to the edge of the crystal substrate 31 and connects to a pad electrode 35a. Further an electrode 33b is formed in the cavity 32 facing against to the electrode 33a, wherein a lead electrode 34b extends from the electrode 33b to the edge of the crystal substrate and connects to a pad electrode 35b to form a high-frequency resonator 30. It has been known that the resonant frequency of a high-frequency crystal resonator 30 is inversely proportional to the thickness of the vibrate portion of the cavity 32, and the levelness and the flatness of the cavity 32 are known to have great influences on the various characteristics of the high-frequency crystal resonator 30 and the spurious output near the resonance frequency.
When a high-frequency crystal resonator is used in a voltage-controlled crystal oscillator (VCXO), it is preferable to drive the resonator on the fundamental mode in order to widen the variable range of the frequency and not to deteriorate the capacitance ratio of the crystal resonator.
FIG. 4 shows another conventional high-frequency crystal resonator 30′ that has been improved to suppress spurious appearing near the resonance frequency of the high-frequency crystal resonator shown in FIG. 3. A pair of secondary electrodes 36a and 36b are formed with a gap around a pair of main electrodes 33a and 33b. The secondary electrodes 36a and 36b are short-circuited each other, and may be grounded so that the shield effect between input and output terminals are brought about by the secondary electrode 36a and 36b. 
In these high-frequency vibratos 30′, it was more difficult to measure various constants accurately at higher frequencies because the various constants are measured by a method using a π circuit through the pair of electrode pads 35a, 35b connecting with the lead electrodes 34a, 34b, and the extending from the main electrodes 33a, 33b. The measurements are prone to the influence of floating capacitance and the like. For example, the IEC standards for π circuit measurements set the upper limit of the measurement to be 125 MHz, and it is not possible to precisely measure for higher frequencies beyond that upper limit.